At first glance, the inverse relationship between interest rates and bond prices seems somewhat illogical, but upon closer examination, makes sense. An easy way to grasp why bond prices move opposite to interest rates is to consider zerocoupon bonds, which don't pay coupons but derive their value from the difference between the purchase price and the par value paid at maturity.
For instance, if a zerocoupon bond is trading at $950 and has a par value of $1,000 (paid at maturity in one year), the bond's rate of return at the present time is approximately 5.26% = (1000950) ÷ 950.
For a person to pay $950 for this bond, he or she must be happy with receiving a 5.26% return. But his or her satisfaction with this return depends on what else is happening in the bond market. Bond investors, like all investors, typically try to get the best return possible. If current interest rates were to rise, giving newly issued bonds a yield of 10%, then the zerocoupon bond yielding 5.26% would not only be less attractive, it wouldn't be in demand at all. Who wants a 5.26% yield when they can get 10%? To attract demand, the price of the preexisting zerocoupon bond would have to decrease enough to match the same return yielded by prevailing interest rates. In this instance, the bond's price would drop from $950 (which gives a 5.26% yield) to $909.09 (which gives a 10% yield).
Now that we have an idea of how a bond's price moves in relation to interestrate changes, it's easy to see why a bond's price would increase if prevailing interest rates were to drop. If rates dropped to 3%, our zerocoupon bond  with its yield of 5.26%  would suddenly look very attractive. More people would buy the bond, which would push the price up until the bond's yield matched the prevailing 3% rate. In this instance, the price of the bond would increase to approximately $970.87. Given this increase in price, you can see why bondholders (the investors selling their bonds) benefit from a decrease in prevailing interest rates.
For example, when the Federal Reserve increased interest rates in March 2017 by a quarter percentage point, the bond market fell. The yield on 30year Treasury bonds dropped to 3.108% from 3.2%, the yield on tenyear Treasury notes fell to 2.509% from 2.575%, and the twoyear notes' yield fell from 1.401% to 1.312%.
For a more detailed explanation of bond pricing and yield calculations, check out Advanced Bond Concepts.

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The amount of return an investor will realize on a bond. Several ... 
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The percentage of par, or face value, at which a bond is quoted. ... 
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Annual income (interest or dividends) divided by the current ... 
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